This is a question from a non-photographer. I'm writing a scene for a stage play. In my scene, there's a celebrity in crisis in a park after sunset and a photographer who'd dearly like to get a picture of her going through her crisis. Lots of flashes would alert her to the photographer's presence. Is it possible to take pictures without flashes of light? Would there be a particular place that the celebrity would need to stand in order for the picture to be possible--like under a streetlight? Would the photographer have to be under a streetlight? I don't need to know what equipment would do this really, just how I'd need to set up the scene in the stage directions, if indeed it is possible.
A streetlight in the park is sufficient for a photograph with modern cameras. However, the photographer doesn't need to be under the streetlight — the subject does. The photographer can (and, given the situation, sounds like should) stay completely in the dark.
Although it's slightly technical, take a look at this chart of exposure values at Wikipeda. In part:
Night street scenes and window displays 7–8 Night vehicle traffic 5
If your city park is well-lit by streetlights, it falls in this range. Let's assume the darker end of the range. This would mean that with an f/2.8 lens and ISO 3200, you would be able to take a photo with a shutter speed of ¹⁄₁₂₅th of a second — fast enough that subject motion blur won't be a big problem.
Any DSLR with a decent lens will be able to do that. The kit lens and cheap zooms won't have as fast an aperture, but if new enough the camera will have higher ISO to compensate. A really cheap point and shoot might struggle, but a decent one will be well within this.
If your character is a photographer with pro-level gear, this isn't even a problem. If you can give your photographer a really big telephoto lens, that should look impressive enough that you shouldn't need to explain that it would work for the pedants in the audience.
If the subject is standing in a streetlight and a car is going by to provide frontal illumination, that would work pretty well. A streetlight could also work, but would have heavy shadows. This could work well for a shot of someone in crisis though.
Honestly, the best bet is to try it out. Get a friend and a camera, go out and take some photos. This is a recent test I did with my 5D Mark iii and a f2.8 lens to see what kind of low light shots I could get. 1,2,6,7,8 and 11 use a flash, but the rest are natural lighting.
Oh, you also would NOT want the photographer under a streetlight. It wouldn't necessarily matter that much, but peripheral light bleed can cause issues for low light exposures.
You might be able to get some inspirations for the stage setup from this review of Rokinon/Samyang 85mm 1.4 - a "fast prime" - where he takes a lot of shots of random people who mostly dont know he is taking them on black friday:
Note also when he says that it seems people who do notice him don't even realize it is taking a photo of them up close, because it looks small like a lens that takes the entire scenery.